For example, some authors require that an attractor have positive measure ( preventing a point from being an attractor ), others relax the requirement that B ( A ) be a neighborhood.

When these sets ( or the motions on them ), are harder to describe than the classical geometric objects, then the attractor is a strange attractor, as described in the section below.

* Georgi Yordanov Georgiev 2012, A quantitative measure, mechanism and attractor for self-organization in networked complex systems, in Lecture Notes in Computer Science ( LNCS 7166 ), F. A.

As such it will leave behind all non-attractor states ( the attractor's basin ), and thus select the attractor out of all others.

Connectivity is the control parameter in the meta learning model developed and empirically validated by Losada ( 1999 ), who found that high performance teams have dynamics that correspond to a complexor ( complex order ) which is mathematically equivalent to a chaotic attractor, representing the flexibility and creativity of these teams ; medium performance teams have dynamics that correspond to a transient limit cycle that eventually settles into a fixed-point attractor, representing the inability to escape limiting routines ; and low performance teams have dynamics that correspond to a fixed-point attractor, representing even less flexibility, and leading to a dead-end situation from which it's very hard to escape.

Assume that the dynamics f has a strange attractor A with box counting dimension d < sub > A </ sub >.

If α < sub > 1 </ sub > = 0. 852 then the real part of one of the complex eigenvalue pair becomes positive and there is a strange attractor.

In the special case where all n < sub > i </ sub > have the same value n, so that the inverse system is determined by the multiplication by n self map of the circle, solenoids were first introduced by Vietoris for n = 2 and by van Dantzig for an arbitrary n. Such a solenoid arises as a one-dimensional expanding attractor, or Smale – Williams attractor, and forms an important example in the theory of hyperbolic dynamical systems.

Whenever x < sub > 0 </ sub > belongs to the attractor of the IFS, all iterations x < sub > k </ sub > stay inside the attractor and, with probability 1, form a dense set in the latter.

and closed oscillations can occur for n > 8 and behave limit cycles: after a perturbation of the system's state, it returns to its previous attractor.

Hénon attractor for a = 1. 4 and b = 0. 3

Hénon attractor for a = 1. 4 and b = 0. 3

Numerical estimates yield a correlation dimension of 1. 25 ± 0. 02 and a Hausdorff dimension of 1. 261 ± 0. 003 for the attractor of the canonical map.

The Kaplan – Yorke dimension, a measure of the dimensionality of the attractor, is 2. 074.

Tinkerbell attractor with a = 0. 9, b =- 0. 6013, c = 2, d = 0. 5.

Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor.

It is seen as an important attractor of tourists to Tasmania and has come to worldwide attention through the Looney Tunes character of the same name.

variables is finite, the attractor distribution is the normal distribution.

Constitutive heterochromatin is usually repetitive and forms structural functions such as centromeres or telomeres, in addition to acting as an attractor for other gene-expression or repression signals.

All finite initial configurations tested eventually converge to the same repetitive pattern, suggesting that the " highway " is an attractor of Langton's ant, but no one has been able to prove that this is true for all such initial configurations.

The choice of starting point may determine which attractor the system ends up on, if there is more than one.

Kepler's laws of planetary motion may be derived from Newton's laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor.

An attractor is a set towards which a variable, moving according to the dictates of a dynamical system, evolves over time.

That is, points that get close enough to the attractor remain close even if slightly disturbed.

The attractor is a region in n-dimensional space.

If the evolving variable is two-or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, ( as for example in the three-dimensional case depicted to the right ).

If the variable is a scalar, the attractor is a subset of the real number line.

If a set of points is periodic or chaotic, but the flow in the neighbourhood is away from the set, the set is not an attractor, but instead is called a repeller ( or repellor ).

This one part of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.

Because of the dissipation, the point is also an attractor.

An attractor is a subset A of the phase space characterized by the following three conditions:

This basically says that the attractor cannot have dimension greater than N-1.

If the derivative is less than zero everywhere except the equilibrium point, then the equilibrium point is a stable fixed point attractor.

Now, instead of having to integrate the system over thousands of time steps to see if any dynamics other than a fixed point attractor exist, one need only determine if the Lyapunov function exists ( note: the absence of the Lyapunov function doesn't guarantee a limit cycle, torus, or chaos ).

( In the 1970s Ruelle and the Dutch mathematician Floris Takens produced a new model for turbulence, and it was Ruelle who invented the concept of a strange attractor in a dynamical system.

The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.

For example, the Ruelle – Takens scenario describes how a periodic orbit bifurcates into a torus and the torus into a strange attractor.

In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations, but which may be combined into a single ( although rather complicated ) jerk equation.

When neurons that are intrinsically rhythmic are connected to each other by excitatory or inhibitory synapses, the resulting networks are capable of a wide variety of dynamical behaviors, including attractor dynamics, periodicity, and even chaos.

) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions.

( However, we will usually be interested in the attractor ( or attractors ) of a dynamical system, and there will normally be one set of exponents associated with each attractor.

An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor.

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor.